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Understanding the
behavior
ofTWO
fluids
UNIT
and
Fluids and
thermodynamics is
Heat
crucial to
understanding
engines and energy
utilization.
Chapter 9
The Behavior of Fluids
Pressure
explains...
floating
objects and
moving
fluids
Pressure and Pascal’s
Principle
Why does a small
woman wearing
high-heel shoes sink
into soft ground
more than a large
man wearing large
shoes?
Pressure
 The man weighs more, so he
exerts a larger force on the ground.
 The woman weighs less, but the
force she exerts on the ground is
spread over a much smaller area.
 Pressure takes into account both
force and the area over which the
force is applied.



Pressure is the ratio of the force to the
area over which it is applied:
N/m2
P
F
A
Units: 1
= 1 Pa (pascal)
Pressure is the quantity that
determines whether the soil will yield.
Pascal’s Principle
 What happens inside a
fluid when pressure is
exerted on it?
 Does pressure have a
direction?
 Does it transmit a force to
the walls or bottom of a
container?
Pascal’s Principle
 Fluid pushes outward uniformly in all
directions when compressed.
 Any increase in pressure is transmitted
uniformly throughout the fluid.
 Pressure exerted on a piston extends
uniformly throughout the fluid, causing it
to push outward with equal force per
unit area on the walls and the bottom of
the cylinder.
 This is the basis of Pascal’s Principle:
 Any change in the pressure of a fluid
is transmitted uniformly in all
directions throughout the fluid.
How does a
hydraulic jack
work?
 A force applied to a piston with a small area can produce a
large increase in pressure in the fluid because of the small area
of the piston.
 This increase in pressure is transmitted through the fluid to the
piston with the larger area.
 The force exerted on the larger piston is proportional to the
area of the piston: F = PA.
 Applying the same pressure to the larger area of the second
piston results in a larger force on the second piston.
A force of 10 N is
applied to a circular
piston with an area of 2
cm2 in a hydraulic jack.
The output piston for
the jack has an area of
100 cm2. What is the
pressure in the fluid?
a)
b)
c)
d)
0.002 Pa
5 Pa
10 Pa
50 kPa
F1 = 10 N
A1 = 2 cm2 = 0.0002 m2
P = F1 / A1 = 10 N / 0.0002 m2
= 50,000 N/m2
= 50 kPa
What is the force
exerted on the
output piston by the
fluid?
a)
b)
c)
d)
50 N
500 N
5,000 N
50,000 N
P = 50 kPa
A2 = 100 cm2 = 0.01 m2
F1 = PA1 = (50,000 N/m2)(0.01 m2)= 500
N
The mechanical advantage is
500 N / 10 N = 50.
Atmospheric Pressure and
the Behavior of Gases
 Living on the surface of the earth, we are at the
bottom of a sea of air.
 This sea of air is thinner at higher altitudes.
 It is also thinner during certain weather
conditions.
 We describe this property by atmospheric
pressure: the pressure of the layer of air that
surrounds the earth.

At sea level, the atmospheric pressure is 100 kPa, or
14.7 pounds per square inch, but it decreases with
altitude.
 Torricelli invented the barometer, a
device for measuring atmospheric
pressure, in an attempt to explain why
water pumps could pump water to a
height of only 32 feet.
 He filled a tube with mercury and
inverted it into an open container of
mercury.
 Mercury worked well because it is much
denser than water.

Density is the mass of an object divided by
its volume.
 Air pressure acting on the mercury in
the dish supported a column of
mercury, of height proportional to the
atmospheric pressure.
 Otto von Guericke performed a famous experiment
to demonstrate the effects of air pressure.
 He designed two bronze hemispheres that could be
smoothly joined together at their rims.
 He pumped the air out of the sphere formed from
the two hemispheres.
 Two eight-horse
teams were unable
to pull the
hemispheres apart.
 In other experiments on variations in atmospheric
pressure, Pascal sent his brother-in-law to the top
of a mountain with a barometer and a partially
inflated balloon.
 The balloon expanded as the climbers gained
elevation.
 This was evidence of a
decrease in the external
atmospheric pressure.
Boyle’s Law
 Variations in the volume
and density of a gas that
accompanies changes in
pressure were studied
by Boyle and Mariotte.
 The density of a column
of air decreases as
altitude increases
because air expands as
pressure decreases.
Boyle’s Law
 Boyle discovered that the volume




of a gas is inversely proportional
to the pressure.
Boyle’s Law: PV = constant
If the pressure increases, the
volume decreases.
The density of a column of air
decreases as altitude increases
because air expands as pressure
decreases.
P1V1 = P2V2
A fixed quantity of gas is held in a cylinder capped at one end
by a movable piston. The pressure of the gas is initially 1
atmosphere (101 kPa) and the volume is initially 0.3 m3. What
is the final volume of the gas if the pressure is increased to 3
atmospheres at constant temperature?
a)
0.1 m3
b)
0.3 m3
c)
1 m3
3 m3
d)
P1 = 1 atm
V1 = 0.3 m3
P2 = 3 atm
V2 = ?
V2 = P1V1 / P2
= (1 atm)(0.3 m3) / 3 atm
= 0.1 m3
Archimedes’ Principle
 The average density of an object compared to a fluid
determines whether the object will sink or float in that liquid.
 The upward force that pushes objects back toward the surface
in liquids is called the buoyant force.
 Archimedes’ Principle: The buoyant force acting on an
object fully or partially submerged in a fluid is equal to the
weight of the fluid displaced by the object.
 The source of the buoyant force is the increase in
pressure that occurs with increasing depth in a fluid.



Atmospheric pressure is greater near the surface of the
earth than at higher altitudes.
The pressure near the bottom of a pool is larger than near
the surface.
The weight of the fluid above contributes to the pressure
that we experience.
Water emerging from a hole near
the bottom of a can filled with
water has a larger horizontal
velocity than water emerging
from a hole near the top.
Weight mg Vdg
Volume Ah
Excess Pressure P
W
A
dgAh
a
dgh
 For example, consider a block submerged in water,
suspended from a string.




The pressure of the water pushes on the block from all sides.
Because the pressure increases with depth, the pressure at the bottom
of the block is greater than at the top.
There is a larger force (F = PA) pushing up at the bottom than there is
pushing down at the top.
The difference between these two forces is the buoyant force.
The buoyant force is proportional
to both the height and the crosssectional area of the block, and
thus to its volume.
The volume of the fluid displaced is
directly related to the weight of
the fluid displaced.
 What forces act on a floating object?
 For the block shown, the weight W is balanced by the string’s tension T
and the buoyant force.
 If there are no strings attached or other forces pushing or pulling on the
object, only the weight of the object and the buoyant force determine
what happens.
 The weight is proportional to the density and volume of the object, and
the buoyant force depends on the density of the fluid and the volume of
the fluid displaced by the object.
What happens if the density of the
object is:
greater than that of the fluid?
less than that of the fluid?
the same as that of the fluid?
Fluids in Motion
 The flow of a fluid is affected by many factors,
including the viscosity of the fluid, a measure of
the frictional effects within the fluid.


The larger the viscosity, the larger the frictional forces
between different layers of the fluid.
Molasses has a larger viscosity than water.
 Size also has an effect; for example, a stream’s
current is faster where the stream is narrow.


Rate of flow, for example of water through a stream or
pipe, is volume divided by time.
Gallons per minute; liters per second; cubic meters per
second.
Fluids in Motion
 The volume of a portion of water of length L flowing
past some point in a pipe is the product of the
length times the cross-sectional area A, or LA.
 The rate at which water moves through the pipe is
this volume divided by time: LA / t.
 Since L / t = v, the rate of flow = vA.
 If the flow is continuous, the rate of flow must be
the same at any point along the pipe.
 If the cross-sectional area A decreases, the speed
v must increase to maintain the same rate of flow.
 The speed will also usually be greatest near the
middle of the stream or pipe.
 The fluid can be imagined as flowing in layers.
 Because of frictional or viscous forces, a thin layer
that does not move is usually next to the walls of
the pipe or trough.
 The fluid speed increases as the distance from the
wall increases.
 Each layer moves more slowly than the one above.
 For a fluid with low viscosity, the transition to the
maximum speed occurs over a short distance from
the wall.
 For a fluid with high viscosity, the transition takes
place over a larger distance, and the speed may
vary throughout the pipe or trough.
 Laminar flow is smooth flow, with no eddies or
other disturbances.


The streamlines are roughly parallel.
The speeds of different layers may vary, but one layer
moves smoothly past another.
 Turbulent flow does have eddies and whorls; the
streamlines are no longer parallel.
 Turbulent flow increases the
fluid’s resistance to flowing
through a pipe.
 Higher speeds are more
likely to exhibit turbulent
flow.
 Higher viscosities are less
likely to exhibit turbulent
flow.
 Examples:



Narrowing of a stream
Water from a spigot
Smoke rising from a cigarette
or candle.
 Huge example: the
famous red spot of
Jupiter
Whorls and eddies
can be seen in the
atmospheric gases.
 The giant red spot
is thought to be a
giant and very
stable atmospheric
eddy.

Bernoulli’s Principle
 Simple harmonic motion occurs when the
energy of a system repeatedly changes from
potential energy to kinetic energy and back
again.
Energy added by doing work
to stretch the spring is
transformed back and forth
between potential energy
and kinetic energy.
Bernoulli’s Principle
 How does a large passenger jet manage to get off
the ground?
 What forces keep it in the air?
 How is a ball suspended in mid-air by a leaf blower?
 What happens if we do work on a fluid?


Bernoulli’s principle applies conservation of energy to
the flow of fluids:
The sum of the pressure plus the P 1 dv 2
constant
2
kinetic energy per unit volume of
a flowing fluid must remain constant.
How does
pressure vary
in pipes and
hoses?
 Will the pressure be greatest in the narrow section or the




wide section?
The speed will be greater in the narrow section.
To keep the sum P + 1/2 dv2 constant, the pressure must be
larger where the fluid speed is smaller.
If the speed increases, the pressure decreases. (This goes
against our intuition.)
This can be shown using vertical open pipes as pressure
gauges.

The height of the column of water is proportional to the pressure.
Pressure
decreases with
increasing
speed.
Blowing across the top of a limp piece
of paper causes the paper to rise,
demonstrating Bernoulli’s principle.
How does an
airplane wing
work?
 The shape and tilt of the wing cause the air to move
faster across the top than across the bottom.
 This causes a lower pressure on the top of the wing.
 The pressure difference produces a net upward
force, or lift, acting on the wing.
 When the lift balances the airplane’s weight, the
airplane will fly.
How can a
ball be
suspended in
mid-air?
A ball is suspended in an upwardmoving column of air produced by a
hair dryer. The air pressure is
smallest in the center of the column,
where the air is moving the fastest.
Why does a
curveball
curve?
The whirlpool of air created by the
spin of the ball causes the air to move
more rapidly on one side than the
other. The difference in pressure
produces a force toward the lowerpressure, higher-airspeed side.